Stoichiometry in Producer–Grazer Systems: Linking Energy Flow with Element Cycling

Abstract

All organisms are composed of multiple chemical elements such as carbon, nitrogen and phosphorus. While energy flow and element cycling are two fundamental and unifying principles in ecosystem theory, population models usually ignore the latter. Such models implicitly assume chemical homogeneity of all trophic levels by concentrating on a single constituent, generally an equivalent of energy. In this paper, we examine ramifications of an explicit assumption that both producer and grazer are composed of two essential elements: carbon and phosphorous. Using stoichiometric principles, we construct a two-dimensional Lotka-Volterra type model that incorporates chemical heterogeneity of the first two trophic levels of a food chain. The analysis shows that indirect competition between two populations for phosphorus can shift predator-prey interactions from a (+, −) type to an unusual (−, −) class. This leads to complex dynamics with multiple positive equilibria, where bistability and deterministic extinction of the grazer are possible. We derive simple graphical tests for the local stability of all equilibria and show that system dynamics are confined to a bounded region. Numerical simulations supported by qualitative analysis reveal that Rosenzweig's paradox of enrichment holds only in the part of the phase plane where the grazer is energy limited; a new phenomenon, the paradox of energy enrichment, arises in the other part, where the grazer is phosphorus limited. A bifurcation diagram shows that energy enrichment of producer-grazer systems differs radically from nutrient enrichment. Hence, expressing producer-grazer interactions in stoichiometrically realistic terms reveals qualitatively new dynamical behavior.

Predator‐driven elemental cycling: the impact of predation and risk effects on ecosystem stoichiometry

“…Our model differs, however, from previous advances that have explored consumer effects on elemental cycling (e.g., DeAngelis 1992; Loreau 1995, 2010; Loladze et al. 2000; Hall et al. 2007) in several important respects.…”

Competition and stoichiometry: coexistence of two predators on one prey

“…(One can imagine a continuously stirred culture in a uniformly lighted chamber with open top and clear walls.) In the model construction, we follow a course outlined in Loladze et al (2000), which deals with one consumer-one prey interactions.…”

“…K represents a constant carrying capacity that we relate to light in the following way: suppose that we fix light intensity at a certain value, then let the prey (which is a photoautotroph) grow with no consumers but with ample nutrients. The prey density will increase until selfshading ultimately stabilizes it at some value, K: Thus, every K value corresponds to a specific limiting light intensity and we might model the influence of higher light intensity as having the effect of raising K; all else being equal (Loladze et al, 2000). Much more mechanistic and realistic modeling of light limited growth exists (Huisman and Weissing, 1994;Diehl, 2002).…”

“…First, let us consider a simpler two-dimensional case: a single consumer on one prey. This case was rigorously analyzed in Loladze et al (2000) using the method of nullclines. A nullcline (or Zero Net Growth Isoclines (ZNGI), Tilman, 1982) is the set of points (on the phase plane) at which species' growth rate is zero.…”

“…It has been shown both experimentally (Urabe and Sterner, 1996;Sterner et al, 1998;Nelson et al, 2001;Urabe et al, 2002Urabe et al, , 2003 and theoretically (Andersen, 1997;Loladze et al, 2000;Muller et al, 2001), that variable chemical composition (i.e. stoichiometry) of prey can significantly affect predator-prey dynamics.…”